Optimal. Leaf size=130 \[ \frac{x (a d+2 b c)}{2 b \sqrt{c+d x^2} (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}-\frac{3 \sqrt{a} c \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 (b c-a d)^{5/2}} \]
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Rubi [A] time = 0.101606, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {470, 527, 12, 377, 205} \[ \frac{x (a d+2 b c)}{2 b \sqrt{c+d x^2} (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}-\frac{3 \sqrt{a} c \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 (b c-a d)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 470
Rule 527
Rule 12
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx &=\frac{a x}{2 b (b c-a d) \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{\int \frac{a c-2 b c x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{2 b (b c-a d)}\\ &=\frac{(2 b c+a d) x}{2 b (b c-a d)^2 \sqrt{c+d x^2}}+\frac{a x}{2 b (b c-a d) \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{\int \frac{3 a b c^2}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{2 b c (b c-a d)^2}\\ &=\frac{(2 b c+a d) x}{2 b (b c-a d)^2 \sqrt{c+d x^2}}+\frac{a x}{2 b (b c-a d) \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{(3 a c) \int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{2 (b c-a d)^2}\\ &=\frac{(2 b c+a d) x}{2 b (b c-a d)^2 \sqrt{c+d x^2}}+\frac{a x}{2 b (b c-a d) \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{(3 a c) \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{2 (b c-a d)^2}\\ &=\frac{(2 b c+a d) x}{2 b (b c-a d)^2 \sqrt{c+d x^2}}+\frac{a x}{2 b (b c-a d) \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{3 \sqrt{a} c \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 (b c-a d)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0212488, size = 54, normalized size = 0.42 \[ \frac{c x^5 \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{(a d-b c) x^2}{a \left (d x^2+c\right )}\right )}{5 a^2 \left (c+d x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 1498, normalized size = 11.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.82377, size = 1134, normalized size = 8.72 \begin{align*} \left [\frac{3 \,{\left (b c d x^{4} + a c^{2} +{\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} -{\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \,{\left ({\left (2 \, b c + a d\right )} x^{3} + 3 \, a c x\right )} \sqrt{d x^{2} + c}}{8 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}}, \frac{3 \,{\left (b c d x^{4} + a c^{2} +{\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt{\frac{a}{b c - a d}} \arctan \left (-\frac{{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt{d x^{2} + c} \sqrt{\frac{a}{b c - a d}}}{2 \,{\left (a d x^{3} + a c x\right )}}\right ) + 2 \,{\left ({\left (2 \, b c + a d\right )} x^{3} + 3 \, a c x\right )} \sqrt{d x^{2} + c}}{4 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 5.24698, size = 402, normalized size = 3.09 \begin{align*} \frac{3 \, a c \sqrt{d} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b c d - a^{2} d^{2}}} + \frac{c x}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{d x^{2} + c}} - \frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c \sqrt{d} - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} d^{\frac{3}{2}} - a b c^{2} \sqrt{d}}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c + 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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